Family of Lines: Examples

In this lesson, I’ll cover a few examples to illustrate the concept of Family of Lines.

Example 1 Find the equation of the line which passes through the point of intersection of L1:2x – 3y + 5 = 0 and L2: 3x – 2y + 5 = 0, and also passes through the origin.Solution Here’s one way to solve the problem.
(i) Find the point of intersection L1 and L2: This point will be (-1, 1)
(ii) Use the two point form to find the required equation: (y – 0)=[(1-0)/(-1-0)](x – 0) or x + y = 0.

But that’s a boring method. I’m interested in family of lines. So here’s another method.

As discussed before, equation of any line passing through the point of intersection of the lines L1=0 and L2 =0 will be of the form L1+λL2 =0, where λ is a parameter.

That is, the required line will be of the form (2x – 3y + 5)+λ(3x – 2y + 5) = 0. That’s half the work done! But what about λ?

Another condition is given to us: The line passes through the origin as well. We can therefore substitute the coordinates to obtain the value of λ.

We have, 2(0)-3(0)+5 + λ(3(0)-2(0)+5) = 0, or λ=-1. And we’re done!

The required line is (2x – 3y + 5)+(-1)(3x – 2y + 5) = 0, or x + y = 0. We didn’t even have to find the point of intersection of the given lines. Saved some effort.

So remember, any line passing through the point of intersection of two given lines will be given by L1+λL2 =0, and the value of the parameter λ will be determined by another given condition.

Example 2 Show that, for all real values of ‘a’, the line ax + 3y + a – 6 = 0 passed through a fixed point. Also find the coordinates of that point.

Solution Hmm.. what about this one?

Now, whenever you’re asked to prove that a line passes through a fixed point, think of family of lines.

The given line can be written as 3y – 6 + a(x + 1) = 0. This looks familiar.. of the form L1+λL2 =0, where L1 is 3y – 6 and L2 is x + 1, and a is the parameter λ.

Well? This represents a family of lines passing through the point of intersection of L1 = 0 and L2 = 0, or 3y – 6 =0, and x + 1 = 0, which is (-1, 2).

Therefore, for all real values of a, the line ax + 3y + a – 6 passes through the (fixed) point (-1, 2). You may substitute the coordinates and verify this yourself.

Example 3 If a + b + c = 0, then show that the line ax + by + c = 0 will always pass through a fixed point. Find the coordinates of that point.

Solution This problem is quite similar to the previous one.

Using the given condition, the line can be written as ax + by – a – b = 0 or a(x – 1)+b(y – 1) = 0 or (x – 1) + (b/a)(y – 1) = 0.
Again, something of the form L1 + λL2 = 0. Therefore, the given line will pass through the point of intersection of the lines x – 1 = 0 and y – 1 = 0, which is (1, 1).

I’ll stop here. Hope you’ve got the idea of the application of family of lines.